Let \(G^{\prime } = (V=W[0,s-1],E)\)  be the graph with edges \(E = \lbrace \lbrace w,w^{\prime }\rbrace  \mid w,w^{\prime } \in W[0,s-1]\rbrace \) .
Local correlation (expectation over the edges) on this
graph \(G^{\prime }\)  is the same as global correlation (expectation over two independent
copies of vertices). Then, we obtain See [1]} or [2]} for the details.
\(\Phi _m -\Phi _{m + 1} \ge \operatornamewithlimits{\mathbb {P}}_{S \sim \Pi _m, \sigma \sim \lbrace \mathbf {Z}_S\rbrace }\left[D(S,\sigma ) \ge \mu _m/2 \right] \cdot \frac{\mu ^2}{2 \cdot 2^{2s}}.\) 
